where \(V_{f(R)}(\Phi ) = \Phi R(\Phi )-f(R(\Phi )),\, \ \Phi =f^{\prime }(R)\)  There is no one-to-one correspondence between the functions \(f(R)\)  and \(V_{f(R)}(\Phi )\) . In fact \(V_{f(R)}(\Phi )=\int g(\Phi )d\Phi \)  depends on a choice of some (local) inverse function \(R=g(\Phi )\) , where \(f^{\prime }(g(\Phi ))=\Phi \) , cf. Appendix A in [1]}..
The four functions of the scalar field we identified at the beginning of the paper, present in action (REF ), are:
\({\mathcal {A}}(\Phi ) = \Phi , \quad {\mathcal {B}}(\Phi ) = 0, \quad {\mathcal {V}}(\Phi ) = V_{f(R)}(\Phi ),\quad \alpha (\Phi ) = 0.\) 
